periodic traveling waves - Institut Camille Jordan

SPECTRAL VALIDATION OF THE WHITHAM EQUATIONS FOR
PERIODIC WAVES OF LATTICE DYNAMICAL SYSTEMS
˘
BUGRA
KABIL AND L.MIGUEL RODRIGUES
Abstract. In the present contribution we investigate some features of dynamical lattice systems near periodic traveling waves. First, following the formal averaging method of Whitham,
we derive modulation systems expected to drive at main order the time evolution of slowly modulated wavetrains. Then, for waves whose period is commensurable to the lattice, we prove that
the formally-derived first-order averaged system must be at least weakly hyperbolic if the background waves are to be spectrally stable, and, when weak hyperbolicity is met, the characteristic
velocities of the modulation system provide group velocities of the original system. Historically,
for dynamical evolutions obeying partial differential equations, this has been proved, according
to increasing level of algebraic complexity, first for systems of reaction-diffusion type, then for
generic systems of balance laws, at last for Hamiltonian systems. Here, for their semi-discrete
counterparts, we give at once simultaneous proofs for all these cases. Our main analytical tool
is the discrete Bloch transform, a discrete analogous to the continuous Bloch transform. Nevertheless, we needed to overcome the absence of genuine space-translation invariance, a key
ingredient of continuous analyses.
Keywords: periodic traveling waves; Whitham averaging; modulation systems; lattice dynamical systems; spectral stability ; discrete Bloch transform.
2010 MSC: 34K13, 34K31, 35B10, 35B27, 35B35, 35P05, 37K60, 37L60, 39A23, 39A30.
Contents
1. Introduction
2. Formal derivation of averaged equations
2.1. Reaction-diffusion case
2.2. General mixed case
2.3. Hamiltonian case
3. Analytic framework
3.1. Integral transform
3.2. Linear evolution
3.3. Algebraic interplay
4. Spectral validation of averaged equations
4.1. Reaction-diffusion case
4.2. General mixed case
4.3. Hamiltonian case
5. Conclusions and Remarks
References
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1. Introduction
A common strategy to tackle the analysis of large time behavior of dynamical systems focuses
on coherent structures, often playing the role of organizing centers for the time evolution. We
Date: April 18, 2015.
Research of L.M.R. was partially supported by the ANR project BoND ANR-13-BS01-0009-01.
1
follow here this line of investigation for lattice dynamical systems and restrict our attention to
periodic traveling waves.
By definition, evolution of lattice dynamical systems is continuous in time but discrete in
space. We consider here particular instances of those where the lattice is one-dimensional and
thus may be assumed to be Z, and the evolution obeys a differential equation
d
U (t) = F(U (t))
dt
where the unknown U associates to any time t ∈ R an Rd -valued sequence U (t) ∈ (Rd )Z
(d ∈ N∗ being a fixed dimension) and F is a vector-field on (Rd )Z that preserves the linear
space of finitely-supported sequences and acts smoothly on it. Alternatively one may view the
evolution as given by an infinite number of differential equations on Rd -valued functions coupled
by a finite-range interaction. We expect that our results could actually deal with nonlocalized
interaction satisfying certain short-range decaying assumptions but we choose to ignore those
here to keep technicalities as low as possible.
There are at least two ways in which such systems arise. They may come directly from
modeling at a discrete level, as for neural networks, granular media, crystals, biological molecules,
optical waveguides, chains of coupled oscillators... Alternatively they may emerge from the
discretization in space — often called semi-discretization — of systems of partial differential
equations, that generate fully continuous dynamical systems, In the latter case, mark that
although a study of dynamical evolutions generated by fully discrete schemes including boundary
conditions would be of much more direct practical use, such a study appears as a daunting task
and it is a common belief that still relevant pieces of information are obtained from investigation
of extended semi-discretized schemes. The reader looking for panoramas on lattice dynamical
systems is referred to [CMP95, CMPVV96, MP96, Pan05, Kev11]. Specially designed technical
tools of wide application range may be found in [Rus89a, Rus89b, CMPS98, MP99a, MP99b,
Ioo00, KK01, CL07, Jam03, FP99, FP02, FP04a, FP04b, HVL07, HVL08, HVL09]. Especially
the latter references include key-ingredients to prove existence of traveling waves for various
class of systems, a fact that is taken as an assumption in the present piece of work.
We are interested here in a particular area of the dynamics, namely what occurs in neighborhoods of periodic traveling waves. Compared to other coherent structures of lattice dynamical systems — such as fronts, kinks, pulses, shocks, solitary waves or breathers —, periodic waves have received relatively less attention. In particular, while the stability analysis of some of the former patterns seems to have reached some maturity — see for instance
[MP99b, FP99, FP02, FP04a, FP04b, BGHR03, BHSZ10, HS13] —, the authors of the present
paper are not aware of even a single example of a comprehensive stability study for periodic
waves1 of some lattice dynamical system. Nevertheless, the interested reader may benefit from
looking at [Kee80, TT95, BR97, WZ97, FV99, PP00, IK00, FR02, Pan05, Car05, FR05, Bak07,
GLR09, Mak11, VZ11, Her12, Jam12, BP13, LHC14], that are at least partly dedicated to
periodic waves of lattice dynamical systems, and mostly focusing on proving their existence.
To set our precise framework, let us specify that we say that, for systems as above, a traveling
wave, sometimes called a uniformly sliding solution, — that is, a solution U with special form
U (t) = (u(j − ct))j∈Z for some speed c ∈ R and some profile u : R → Rd — is said to be
periodic if the corresponding profile u is itself periodic. In this case we rescale phase variable
of the profile to ensure that the profile period is one. Explicitly any periodic traveling wave
U is written as U (t) = (u(k j + ω t))j∈Z with a one-periodic profile u. This brings out spatial
wavenumber k ∈ R∗ and time frequency ω = −k c. Mark that up to now we are not assuming
that the spatial wavenumber k is rational. When its is not the case the spatial periodicity of
the profile is not a priori easily observed on the solution itself, though when standing waves are
discarded, that is when restricting to c 6= 0 this periodicity is apparent in time, the solution
being 1/ω-periodic in time. We refer the reader to [Her12] for further elaboration on this
1In a sense analogous to [Sch96, Sch98, MSU01, JNRZ14, Rod13] for dissipative continuous systems.
2
comment including fascinating illustrating pictures. This already stresses that, in contrast with
analogous issues for continuous dynamical systems, analysis of the behavior near standing waves
follows from different kind of arguments. Indeed, while profile equations for genuinely-traveling
waves are differential functional equations, profile equations for standing waves are algebraic
functional equations. This results in the fact that even when traveling and standing waves
coexist the standing-wave limit c → 0 is very singular. For this reason, our analysis focused on
on genuinely-traveling waves do not extend in a straightforward way neither to breathers nor to
steady spatially periodic solutions.
Our goals are, first, to derive — on a formal basis — a system of partial differential equations
that is expected to describe the time evolution of slowly modulated wavetrains of lattice dynamical systems, second, to elucidate — on mathematical grounds — what are, at the spectral
level, the connections between the linearization about a constant state of the formally-derived
modulation systems and the linearization about a periodic wave of our initial lattice system of
equations. Our formal derivation is in many ways very classical and it is well-known as one of
the possibilities to recover a continuous description from a discrete model. The reader is referred
to [HLM94, HLM95, DHM06, DHR06, GHM06, DH08] for various perspectives on this point
of view. Our main motivation here is to provide a rigorous spectral validation, that should be
compared with some contributions of [DSSS09, Ser05, NR13, BGNR14] on partial differential
equations, essentially as [FP99, FP02, FP04a, FP04b] may be thought as a discrete counterpart
of [PW92, PW94].
As a direct consequence of our results stems the proof that characteristic speeds of averaged
systems do provide group velocities for the original one. Mark that even if we illustrate our
strategy mostly on systems obtained from semi-discretization of continuous models we provide a
proof at the discretized level so that, in these particular cases, group velocities implicitly depend
on the mesh size and of the particular choice of discretization. Given the fundamental role of
a precise knowledge of group velocities in a quantitative description of the dynamics, and since
accurate reproduction of dispersion relations obtained by linearizing about constant states has
served for a long time now to discriminate between numerical schemes with, from other respects,
similar performances, we do not exclude that in a near future comparison between discrete group
velocities and continuous ones could serve similar purposes.
Our contribution may seem relatively modest in that the proved connection is purely at the
spectral level. Yet on the other hand our strategy is very robust as we emphasize by treating
here in a single place various classes systems that were considered at the continuous level in
separate contributions. Moreover, as our proofs provide relations for eigenmodes — including
eigenvectors and not restricting to eigenvalues —, we expect that it could be directly useful
for a dynamical validation of averaged systems in the large-time limit, as, in continuous cases,
[DSSS09, Section 4.2] for [SSSU12, JNRZ13], [NR13] for [JNRZ14] or [BGNR14] for [Rod].
From our point of view, our main restriction is that in the spectral validation we only consider
periodic waves whose period is a multiple of the generator of the lattice. This stringent restriction
does not fit well in the spirit of slowly modulated wavetrains that involve a continuum of such
periodic waves but it enables us to use techniques originating in the study of differential operators
with periodic coefficients — the classical Floquet theory — rather than those for quasi-periodic
coefficients. For a glimpse at the fascinating but technically daunting quasi-periodic theory the
reader is referred to [Eli97, Eli02, Eli98, Eli99] where is analyzed the spectrum of operators
similar to those obtained by linearizing discrete equations about a general spatially-periodic
stationary solution.
From a technical point of view, in our analysis of lattice systems, departures from the continuous strategy originate in the loss of a genuine space-translation invariance of the original
system — see Remark 3.1 though. To balance this, we make an extensive use of the remaining time-translation invariance and of the space-translation invariance of the profile equation.
In particular, instead of moving to a mobile frame in which the wave is at rest and analyzing
the spectrum of the generator of the linearized dynamics, we study directly Floquet multipliers
3
of the linearized time evolution, that is, we study the spectrum of the map that encodes the
evolution over a time 1/ω according to the linearized dynamics. Likewise, when dealing with
Hamiltonian systems, we replace in our arguments the missing momentum conservation law with
local conservation of energy. On the other hand, we may still use a normal, Floquet exponent
by Floquet exponent, Bloch-wave decomposition by relying upon the discrete Bloch transform.
Then to examine the spectrum of resulting discrete Bloch symbols we perform here a direct
spectral perturbation analysis ”`
a la Kato” [Kat76] as in [NR13, BGNR14] for continuous systems. Alternatively we could also have introduced a suitable Evans function, as done in [Ser05]
for continuous dynamics and in [KK01, BG02, CL07] for other classes of coherent structures in
lattice systems. For a detailed discussion of respective advantages of proofs by direct spectral
perturbation or by Evans function expansions see [Rod13].
The rest of the present paper is organized as follows. In the second section, examining
formal expansions, we derive modulated equations. In the third section we introduce tools from
functional analysis — including the discrete Bloch transform — that we shall subsequently use
to analyze the spectrum of the linear one-period evolution operator. In Section 4 we state and
prove our main results concerning the spectral validation of the previously obtained averaged
systems. We close this paper with further comments and open questions. In Sections 2 & 4, we
analyze three classes of systems in increasing order of algebraic complexity. First, we consider
a lattice system of reaction-diffusion type, in the sense that the system does not support any
local conservation law. In this case, the averaged evolution obeys a scalar equation and both
derivation and spectral validation are considerably simpler so that we also provide high-order
versions of those. It is actually those higher-order versions that are needed in the dynamical
large-time validation of the slow modulation scenario. After that, we examine a general system
that includes built-in conservation laws. The averaged dynamics is now prescribed by a genuine
system and the spectral validation includes spectral perturbation of a Jordan block, hence
requires a preliminary desingularization. At last, we consider a lattice system of Hamiltonian
type that comes with an ”extra” conservation law encoding conservation of energy. Again, all
the corresponding theorems, Theorem 4.1, Theorem 4.2 and Theorem 4.3, are stated and proved
in Section 4.
Notation. Elements U ∈ (Cd )Z of (Cd )Z are implicitly assumed to have coordinates Uj =
U (j) ∈ Cd , j ∈ Z. Likewise for U ∈ (Cd )Z×R , Uj (t) = U (j, t), for (j, t) ∈ Z × R, and
U (t) = U (·, t) ∈ (Cd )Z , for t ∈ R. All along T denotes the left shift operator on (Cd )Z ,
(T U ) (j) = U (j + 1) ,
for U ∈ (Cd )Z .
The set N is the set of nonnegative integers, and, for (M, N ) ∈ Z, [[M, N ]] = [M, N ] ∩ Z. The
resolvent set and the spectrum set of an operator are denoted respectively by ρ(·) and σ(·).
2. Formal derivation of averaged equations
In this section we derive, from formal considerations, averaged modulation systems for three
classes of nonlinear lattice systems, starting with the simplest case of lattice systems of reactiondiffusion type. To do so, we will follow the two-timing method introduced by Whitham [Whi74].
Only afterwards, in Section 4 , shall we prove that the original dynamical evolution is indeed
related to these modulation equations.
2.1. Reaction-diffusion case. We consider a discretization of the scalar reaction-diffusion
equation for u : R × R+ → Rd
(2.1)
∂t u = ∆u + f (u)
Rd
Rd
where f :
→
is a given (in general nonlinear) smooth function. Using a centered difference
scheme to discretize the spatial Laplacian operator, we obtain the following system of coupled
4
ordinary differential equations
d
Uj (t) = µ[Uj+1 (t) − 2Uj (t) + Uj−1 (t)] + f (Uj (t)),
j ∈ Z, t > 0,
dt
where µ ∈ R is a given fixed constant. To consider it in the abstract form of dynamical lattice
systems, we introduce F the nonlinear operator on U ∈ (Cd )Z associated with f and given by
F (U ) = (f (Uj ))j∈Z . Then, we write system (2.2) for U (·) = (Uj (·))j∈Z as
(2.2)
d
U (t) = µ (T − 2Id + T−1 ) U (t) + F (U (t))),
t > 0.
dt
A profile u generates through U (t) = (u(kj + ω t))j∈Z , a periodic traveling wave to (2.3) with
spatial wavelength k 6= 0 and time frequency ω if and only if
(2.3)
(2.4)
ωu0 (ζ) = µ[ u(ζ + k) − 2 u(ζ) + u(ζ − k) ] + f (u(ζ)),
ζ ∈ R,
and u is one-periodic
(2.5)
ζ ∈ R.
u(ζ + 1) = u(ζ),
To obtain (even formal) pieces of information about a given periodic wave, we need a precise knowledge of neighboring periodic waves. With this respect we make the following ”non
degeneracy” assumption:
(A)
Periodic traveling waves of (2.3) — identified when coinciding up to time
translation — form a smooth manifold of dimension one, regularly parametrized by k.
For latter use, we denote by uk (·), ω(k) such a parametrization by k. We refer the reader to
references given in the introduction for proofs of existence results. We also stress that our assumption is consistent with both the continuous limit leading to (2.1) and similar considerations
in [HLM94, HLM95, DHM06, DHR06, GHM06, DH08].
Our formal ansatz is of two-scale type, one scale slow — but otherwise arbitrary —, involving
macroscopic variables, another one fast but oscillatory, hence described through an oscillation
phase. Explicitly we want to gain some insight about the asymptotic behavior — when the slow
frequency goes to zero — of families of solution (U () (·)) expanding as



φ() (j, t) 
()
 + O(2 ),
Uj (t) = u() 
εj
,
εt
,

|{z} |{z}
| {z }
(2.6)
X
T
Θ
with phases φ()
(2.7)
φ() (X, T ) = φ0 (X, T ) + φ1 (X, T ) + 2 φ2 (X, T ) + O(3 ) ,
and profiles u() one-periodic in the third variable Θ
u() (X, T, Θ) = u0 (X, T, Θ) + u1 (X, T, Θ) + O(2 ) .
This yields the following relation
!
!
() (X, T )
() (X, T )
φ
φ
∂T u() X, T,
+ (∂T φ() (X, T )) · ∂Θ u() X, T,
"
!
!
!#
() (X + , T )
() (X, T )
() (X − , T )
φ
φ
φ
= µ u() X + , T,
− 2u() X, T,
+ u() X − , T,
!!
φ() (X, T )
+ f u() X, T,
+ O(2 ) .
5
Usual identification of powers of and replacement of the phase φ0 (X, T )/ + φ1 (X, T ) with Θ
then provide at order 0
(∂T φ0 (X, T )) ∂Θ u0 (X, T, Θ)
=
µ [ u0 (X, T, Θ + ∂X φ0 (X, T )) − 2u0 (X, T, Θ) + u0 (X, T, Θ − ∂X φ0 (X, T )) ]
+
f (u0 (X, T, Θ)) ,
which is, in the Θ variable, the functional profile equation with wavenumber and time frequency
(2.8)
k0 (X, T ) := ∂X φ0 (X, T )
and
ω0 (X, T ) := ∂T φ0 (X, T ) .
Hence, under a suitable normalization of parametrization, without loss of generality
(2.9)
u0 (X, T, Θ) = uk0 (X,T ) (Θ)
where evolution of the local wavenumber obeys
(2.10)
∂T k0 − ∂X (ω(k0 )) = 0 ,
stemming from Schwarz’ identity ∂T ∂X φ0 = ∂X ∂T φ0 .
As a result of our heuristics, we obtain the conjecture that slow/oscillatory solutions evolve
at main order according to a slow modulation scenario
()
k0 ( j, t) φ0 (j, t)
Uj (t) = u
+ φ1 (j, t) + O(),
that is, locally, at scale 1, the solutions look like one member of the periodic traveling wave family,
but with local parameters — phase shift φ1 and wavenumber k0 — evolving on a slow scale 1/.
In this description local phase and local spatial wavenumber are tied by k0 = ∂X φ0 and the
evolution of the local wavenumber k0 obeys at main order (2.10). In particular, unsurprisingly,
in this simple context, linear group velocity around a given wave U generated by a profile uk is
expected to be ω 0 (k).
In this simple case where the family of periodic waves is one-dimensional and hence modulation
equation (2.10) is scalar, algebra required to raise the order of description remains relatively
simple so that we explain it now. It requires a more precise account of assumption (A). In
assuming a regular parametrization by k, we include that for any wave under consideration —
u with wavenumber k and frequency ω — the operator L acting on L2per ([0, 1]) with domain
1 ([0, 1]) through
Hper
(2.11)
Lu = −ω u0 + µ[ u(· + k) − 2u + u(· − k) ] + df (u(·)) u
has one-dimensional kernel. By time translation invariance, this kernel is thus C u0 . Moreover
since, as a (relatively) compact perturbation of −ω( )0 , L is Fredholm of index 0 — when
considered as usual as a bounded operator on its domain endowed with its graph norm —, we
1
∗
conclude the existence of a uad
k ∈ Hper ([0, 1]) spanning the kernel of the adjoint L of L and such
0
2
that huad
k , u i = 1 (in the Lper ([0, 1]) sense). In particular by differentiating the profile (2.4) and
taking a scalar product with uad
k , one receives
(2.12)
0
0
ω 0 (k) = huad
k , µ[u (· + k) − u (· − k)]i .
Moreover, uad
k allows us to give a precise account of what is the appropriate normalization of
parametrization that we have alluded to above. Indeed we explicitly require
(N)
k
huad
k , ∂k u i = 0
in order to enforce the distinct respective roles of wavenumber and phase shift. The above
condition may be achieved by shifting the original uk in a suitable k-dependent way.
6
After these preliminaries we may go back to the identification of terms of order 1 . By
dropping all (X, T ) dependencies, we receive
−Lu1 + ∂T φ1 ∂Θ u0 − ∂X φ1 µ [∂Θ u0 (· + k0 ) − ∂Θ u0 (· − k0 )]
= −∂T k0 ∂k uk0 + µ∂X k0 [∂k uk0 (· + k0 ) − ∂k uk0 (· − k0 )]
1
+ µ∂X k0 [∂Θ u0 (· + k0 ) + ∂Θ u0 (· − k0 )]
2
where L is here an (X, T )-parametrized version of original L, associated with u0 and acting in
the Θ-variable. Averaging against uad
k0 yields
∂T φ1 − ω 0 (k0 ) k1 = d(k0 ) ∂X k0 .
where, for concision’s sake, we have introduced a diffusion coefficient
(2.13)
k
k
k
k
1
d(k) = huad
k , µ [∂k u (· + k) − ∂k u (· − k)] + 2 µ [∂ζ u (· + k) + ∂ζ u (· − k)]i .
As a refined conclusion, we obtain that slow/oscillatory solutions evolve at main order according to
()
k( j, t) φ(j, t)
+ O() ,
Uj (t) = u
with local phase and local spatial wavenumber tied by k = ∂X φ and the evolution of the local
wavenumber k obeys at second order
(2.14)
∂T k − ∂X (ω(k)) = ∂X (d(k)∂X k) .
Mark that in this second-order description, there is no more undetermined local phase shift
and that, when present, large-time diffusive decay or anti-diffusive growth may be captured by
equation (2.14).
2.2. General mixed case. The main point about discrete reaction-diffusion systems considered in the previous subsection is that they do not support any built-in conservation law. Assumption A then requires that indeed no hidden conservation is present. This leads to a scalar
modulation behavior encoded by (2.10) or (2.14). We now relax this stringent assumption to
consider a system mixing conservative equations with non conservative ones.
At the continuous level such structure is ubiquitous. For instance it naturally emerges from
the modeling of the evolution of isentropic compressible flows undergoing external forces by the
Navier-Stokes or Euler systems. Typically, in the former case one receives for the evolution of
mass density ρ and velocity field u a system in the form
(2.15)
∂t ρ + ∂x (ρ u) = 0
∂t (ρu) + ∂x (ρu2 + P (ρ)) = ∂x (ν(ρ)∂x u) + g(ρ, u),
where P , ν and g provide respectively the pressure law, Lam´e viscosity coefficient and external
forces.
Here we consider for U = (R, W )T — valued in (Rd )Z = (Rd1 )Z × (Rd2 )Z — systems of the
following general form
(2.16)
d
R(t) + D1 (fr (R(t), W (t))) = 0
dt
d
W (t) + D2 (fw (R(t), W (t))) = D3 (B(R(t), W (t))D4 W (t)) + g(R(t), W (t)) ,
dt
where f = (fr , fw )T : Rd → Rd , g : Rd → Rd2 and B : Rd → L(Rd2 ) are identified with
their component-wise counterparts actions on sequences, of respective type (Rd )Z → (Rd )Z ,
(Rd )Z → (Rd2 )Z and (Rd )Z → L((Rd2 )Z ), and Dj , j = 1, 2, 3, 4, are constant-coefficient discrete
differential operators, D1 being conservative in the sense that the kernel of D∗1 (the adjoint of
7
D1 on `2 (Z; Rd1 )) contains constant sequences. From now on, for definiteness’ sake, we will
restrict our attention to the case where for some η > 0
D2 = D3 = η (Id − T−1 ) .
D1 = D4 = η (T − Id) ,
(2.17)
As in the reaction-diffusion case, we make a ”non degeneracy” assumption on the set of
periodic traveling waves:
Periodic traveling waves of (2.16) — identified when coinciding up to time
translation — form a smooth manifold of dimension d1 + 1, regularly parametrized
by (k, M), their wavenumber and the average-values of their d1 th first components.
R1
Explicitly M = (M1 , · · · , Md1 ) with Mj = 0 ej · u with ej the jth element of the canonical
basis of Rd . For latter use, we denote by uk,M (·), ω(k, M) a parametrization provided by the
previous assumption. Again we refer the reader to references in the introduction for proofs of
existence results.
Again, a formal ansatz


(A)

φ() (j, t) 
()
 + O(2 ),
Uj (t) = u() 
εj
,
εt
,
|{z} |{z}

| {z }
(2.18)
T
X
with phases
(2.19)
Θ
φ()
φ() (X, T ) = φ0 (X, T ) + φ1 (X, T ) + 2 φ2 (X, T ) + O(3 ) ,
and profiles u() one-periodic in the third variable Θ
u() (X, T, Θ) = u0 (X, T, Θ) + u1 (X, T, Θ) + O(2 )
leads, under a suitable normalization of parametrization, to
u0 (X, T, Θ) = u(k0 ,M0 )(X,T ) (Θ)
with
k0 (X, T ) := ∂X φ0 (X, T ) ,
and
Z
ω0 (X, T ) := ∂T φ0 (X, T )
1
0d1 ×d2 u0 (X, T, Θ) dΘ .
Id1
M0 (X, T ) :=
0
But now the law of conservation of waves
∂T k0 − ∂X (ω(k0 , M0 )) = 0
fails to describe completely the time-evolution.
To proceed, one collects terms of power 1 , replacing φ0 (X, T )/ + φ1 (X, T ) with Θ, and, with
usual implicit notation for coordinates splitting u = (r, w)T , receives
0
−Lu1 + ∂T (u0 ) + ηTk0 (df (u0 ) · ∂X u0 ) = ∂Θ (· · · ) +
,
···
where L is the operator associated with the linearized profile equation and Tk0 acts on functions
of Θ through Tk0 (v) = v(· + k0 ). Motivated by the fact that constant functions with values
ej , j = 1, . . . , d1 , lie in the kernel of L∗ , we average over Θ ∈ (0, 1) the scalar product of the
previous system with those ej and obtain
∂T M0 + ∂X F (k0 , M0 ) = 0
where
Z
1
fr (uk,M (Θ)) dΘ .
F (k, M) = η
0
8
Altogether we have derived the following averaged system

 ∂T k − ∂X ω(k, M) = 0
(2.20)
∂ M + ∂ F (k, M) = 0
T
X
for the evolution of local parameters involved in a slow modulation description of the dynamics.
2.3. Hamiltonian case. Up to now we have implicitly assumed that our systems do not contain
any ”hidden” conservation law. In doing so we were motivated by our will to restrict to nondegenerate cases. However for special algebraic classes of systems ’hidden’ conservation laws are
indeed generic and we need to accommodate them. As a typical example we analyze systems that
are (at least formally) of Hamiltonian type. Incidentally, although we do not follow this path
here, we stress that the presence of an extra structure also offers alternative ways of deriving
the same modulation equations; see [Whi74].
To emphasize analogy with results on continuous dynamical systems in [BGNR14] we focus
on a class of Hamiltonian lattice dynamical systems that include discrete counterparts to the
Korteweg–de Vries equation and the Euler–Korteweg system. Explicitly we fix some Hamiltonian
H : (Rd )2 → R and some skew-symmetric constant-coefficient discrete differential operator J
in a conservative form, say for definiteness’ sake J = D B with B a symmetric matrix of size d
and D = η2 T − T−1 . Moreover we choose some other constant-coefficient discrete differential
e (not necessarily skew-symmetric), say D
e = η (T − Id), for some η > 0.
operator D
Then we consider the following lattice dynamical system
d
U (t) = J δH[U (t)],
dt
where δ denotes a discrete Euler operator providing variational derivatives for functionals steme ),
ming from local functions of (U, DU
(2.21)
(2.22)
e ).
e U) + D
e ∗ ∇ e H(U, DU
δH[U ] = ∇U H(U, D
DU
By definition the previous system comes with a local conservation law for the discrete Hamiltonian
d
−1
1 −1
e
e
e
(2.23)
[H(U, D U )] = D 2 T (δ H[U ]) B δ H[U ] + T (∇D
e U H(U, D U )) · J δ H[U ] .
dt
Notice that to derive (2.23) from (2.21) we have used that
−1
e − (D
e ∗ A) B = D((T
e
A DB
A)B)
for any A, B ∈ (Cd )Z .
As we have already done in previous subsections, to the shift operator T acting on sequences
we may naturally associate a family of shifts on functions Tk (v) = v(· + k). For concision’s
sake, from now on we will likewise denote by (P (T))k the operator P (Tk ). In particular, we
e k , δk ...
will freely use notation Dk , D
Now, our ”non-degeneracy” assumption on the set of periodic traveling waves takes the following form
(A)
Periodic traveling waves of (2.21) — identified when coinciding up to time translation
— form a smooth manifold of dimension d + 2, regularly parametrized by (k, M, E),
their wavenumber, the average-values of their components and their Hamiltonian.
More explicitely
Z
E =
1
e k u(k,M,E) (ζ)) dζ .
H(u(k,M,E) (ζ), D
0
9
With notational convention similar to the one of previous subsections, a two-scale ansatz
leads, under a suitable normalization of parametrization, to
u0 (X, T, Θ) = u(k0 ,M0 ,E0 )(X,T ) (Θ)
with
k0 (X, T ) := ∂X φ0 (X, T ) ,
ω0 (X, T ) := ∂T φ0 (X, T ) ,
1
Z
M0 (X, T ) :=
u0 (X, T, Θ) dΘ ,
0
and
Z
E0 (X, T ) :=
0
1
e k (X,T ) u0 (X, T, Θ)) dΘ .
H(u0 (X, T, Θ), D
0
Moreover we still have a law of conservation of waves
∂T k0 − ∂X (ω(k0 , M0 )) = 0
and a conservation law for the averaged-values
∂T M0 = B ∂X F (k0 , M0 , E0 ),
where
Z
F (k, M, E) = η
1
δk (H)[uk,M,E ](ζ) dζ = η
0
Z
1
e k uk,M,E (ζ)) dζ .
∇U H(uk,M,E (ζ), D
0
The latter fact is easily deduced from the fact that by collecting terms of power 1 stemming
from the insertion of our ansatz in our system we receive an equation in the form
−Lu1 + ∂T (u0 ) − η2 (Tk0 + T−k0 )B∂X (δk0 (H)[u0 ]) = ∂Θ (· · · ) + Dk0 (· · · ) .
The derivation of an equation for the time evolution of E0 requires more algebra. To write
computations in a compact way we introduce some more pieces of notation. First we set Lξ :=
e−iξ· L eiξ· then we denote L(1) and L(2) the operators involved in the expansion
Lξ = L + (iξ k) L(1) + (iξ k)2 L(2) + O(|ξ|3 ) .
With these preliminaries the equation obtained from the collection of coefficients of 1 is explicitly
written as
−Lu1 + ∂T (u0 ) + (∂T φ1 + c(k0 , M0 , E0 )∂X φ1 ) ∂Θ u0
2 φ L(2) ∂ u + (L(1) − c(k , M , E )) ∂ u .
= ∂X φ1 L(1) ∂Θ u0 + ∂X
0
0
0
0
Θ 0
X 0
Since δk0 [H](u0 ) belongs to the kernel of L∗ , we shall average over Θ ∈ (0, 1) the scalar product
of the previous system with δk0 [H](u0 ). Our claim is that we receive
∂T E0 − ∂X (S(k0 , M0 , E0 )) = 0
where
Z
1
1
2
S(k, M, E) = η
0
Z
+ η
0
T−k (δk (H)[u(k,M,E) ]) B δk (H)[u(,M,E) ]
1
(k,M,E) e
T−k (∇D
, Dk u(k,M,E) )) · Jk δk (H)[u(k,M,E) ]
e U H(u
Indeed it does follow from direct computations. One one hand, when a ∈ {ζ, k, M1 , · · · , Md , E},
one computes that
(2.24)
e k u)) − ηT−k (∇ e H(u, D
e k u)) · ∂ζ u ∂a k + (· · · )
δk [H](u) · ∂a u = ∂a (H(u, D
DU
10
where (· · · ) denotes mean-free terms and dependence of u on parameters has been omitted. On
the other hand, for a ∈ {ζ, k, M1 , · · · , Md , E}, we also have that
(2.25)
δk [H](u) ·
(L(1) − c)∂a u + L(2) (∂ζ u)∂a k
e
= ∂a η 21 T−k (δk (H)[u]) B δk (H)[u] + ηT−k (∇D
e U H(u, Dk u)) · Jk δk (H)[u]
e
− ηT−k (∇D
e U H(u, Dk u)) · ∂ζ u ∂a ω + (· · · )
with the same convention. This yields our claim by integration using that ∂T k0 = ∂X (ω(k0 , M0 , E0 )).
Alternatively one could have derived this last equation by directly expanding and averaging
equation (2.23).
Altogether we have derived the following averaged equations for the slow modulation of local
parameters

∂T k = ∂X ω(k, M, E)


∂T M = B∂X F (k, M, E)
(2.26)


∂T E = ∂X S(k, M, E) .
3. Analytic framework
We collect in this section some preliminaries from functional analysis that will allow us to
formulate in which sense the formally derived averaged equations provide some valuable pieces
of information at the spectral level.
3.1. Integral transform. To analyze linearized lattice systems with periodic coefficients we
shall make use of an adapted integral transform, usually called discrete Bloch transform. Though
probably less known that its continuous counterpart it is a natural functional tool that have
already received various applications in related contexts, see for instance [CBCPS12, CBS12].
3.1.1. Discrete Bloch Transform. Let N ∈ N∗ be given. We define the following transform
π π d
(3.1)
B : `2 (Z; Cd ) → L2 [[0, N − 1]] × − N
, N ; C , f 7→ fˇ
where for j ∈ [[0, N − 1]]
(3.2)
X
fˇ(j, ·) := lim
k0 →∞
e−i(kN +j) · fkN +j ,
k∈[[−k0 ,k0 ]]
π π
the limit being taken in the L2 − N
, N -sense. Up to a multiplicative constant this is a total
isometry whose reciprocal is given as
π π d
(3.3)
B −1 : L2 [[0, N − 1]] × − N
, N ;C
→ `2 (Z; Cd ) , fˇ 7→ f
where for (k, j) ∈ Z × [[0, N − 1]]
(3.4)
fkN +j
N
=
2π
Z
π
N
π
−N
ei (kN +j) ξ fˇ(j, ξ) dξ .
From now on we’ll constantly identify functions on Z/(N Z) with functions on [[0, N − 1]]
through usual restriction/periodic extension processes. To this purpose we stress that formula (3.2) already provides the needed periodic extension. With this identification, inverse
formula (3.4) simply reads for any j ∈ Z
Z π
N ij ξ
N
(3.5)
fj =
e
fˇ(j, ξ) dξ .
2π − π
N
11
π π
3.1.2. Discrete Bloch symbols. To any element ξ 7→ Aξ of L∞ ( − N
, N ; L(`2 (Z/(N Z); Cd )))
one may associate a bounded operator A ∈ L(`2 (Z; Cd )) such that, for any f ∈ `2 (Z; Cd ),
(Af )ˇ(·, ξ) = Aξ fˇ(·, ξ) .
Explicitly
(3.6)
N
=
2π
(Af )j
Z
π
N
π
−N
ei j ξ (Aξ fˇ(·, ξ))j dξ .
Moreover up to a multiplicative constant the map
π π
, N ; L(`2 (Z/(N Z); Cd ))) → L(`2 (Z; Cd ))
L∞ ( − N
is a partial isometry. From this follows the classical identity between the spectrum of the
synthetized operator A and the
essential range of ξ 7→ σ(Aξ ). In particular, in the case where
π π
ξ 7→ Aξ belongs to C 0 ( − N
, N ; L(`2 (Z/(N Z); Cd ))), this yields
[
σ(Aξ )
σ(A) =
π π
ξ∈[− N , N ]
so that the spectrum of A may be analyzed by looking at the family of spectra of the finitedimensional symbols Aξ .
Reciprocally for a large class of A ∈ L(`2 (Z; Cd )) one may define corresponding symbols
ξ 7→ Aξ formally given by
Aξ = e−i · ξ A ei · ξ
that is, for f ∈ `2 (Z/(N Z); Cd )
(Aξ f )j = e−i j ξ (A (ei · ξ f ))j
with an extended sense of A. To make things more concrete, we focus now on the case where A
is an N -periodic discrete differential operator in the sense that it is given as
X
A = a(·, T) with a(·, X) =
aj (·) X j
j∈Z
L(Cd )
where T is again the left shift operator and a : Z →
of A are then given by
Aξ = a(·, ei ξ T)
is N -periodic. The Bloch symbols
where T is now the left shift operator acting on (Cd )Z/(N Z) and a is considered as a function
on Z/(N Z).
3.2. Linear evolution. By linearizing original equations we receive equations in the form
d
V (t) = A(t) V (t)
dt
where A(·) is a 1/ω-periodic family of bounded operators generating S(·, ·) an evolution system
on `2 (Z; Cd ) (in the sense defined for instance in [Paz83, Chapter 5]). Explicitly we have
(∀s ∈ R, S(s, s) = Id)
and
(∀(s, r, t) ∈ R3 , S(t, s) = S(t, r)S(r, s))
and
∀(s, t) ∈ R2 , (∂t S)(t, s) − A(t) S(t, s) = 0
and
(∂s S)(t, s) + S(t, s) A(s) = 0 .
Mark that periodicity implies that linear stability in the sense that
sup kS(t, 0)k < ∞
t∈R+
is equivalent to
sup kS(1/ω, 0)k k < ∞ .
k∈N
12
Likewise linear exponential instability in the sense that
∃γ > 0 , inf e−γ t kS(t, 0)k > 0
t∈R+
is equivalent to
∃κ > 1 , inf κ−k kS(1/ω, 0)k k > 0 ,
k∈N
that is to ρ(S(1/ω, 0)) > 1.
Remark 3.1. Here we use time-translation invariance and periodicity in time as a substitute
to space-translation invariance and stationarity in a moving frame that play a crucial role in
similar analyses of continuous systems. However as pointed out for instance in [FP99, FP02,
FP04a, FP04b] the loss of an exact invariance in space does not preclude arguments based on a
pseudo-invariance. Notably, mark that for all t ∈ R+
A(t + 1/c) = T−1 A(t)T .
This implies that for all (t, s) ∈ R
S(t + 1/c, s + 1/c) = T−1 S(t, s)T .
As a consequence, linear stability is also equivalent to
sup k(TS(1/c, 0))k k < ∞ .
k∈N
The fact that such an alternative exist is obviously crucial when dealing with fronts, shocks or
solitary waves rather than with periodic waves.
From now on, in order to make the most of the discrete Bloch transform we focus on cases
where, for some N ∈ N∗ , A(·) is valued in N -periodic discrete differential operators that is we
restrict to cases where linearization is about a periodic traveling wave with an integer period, N .
Then Aξ (·) the value at ξ of the corresponding family of Bloch symbols generates the evolution
system given by the value at ξ, Sξ (·, ·), of the family of Bloch symbols associated with S(·, ·).
In our context some continuity in the ξ-variable is available and linear stability reads
sup
sup kSξ (1/ω, 0)k k < ∞ ,
π π
ξ∈[− N , N ] k∈N
while linear exponential instability may be written as
π π
∃ξ ∈ − N
, N , ρ(Sξ (1/ω, 0)) > 1 .
π π
This implies that a necessary condition for linear stability is that for any ξ ∈ − N
,N
¯ 1) .
σ (Sξ (1/ω, 0)) ⊂ B(0,
The fact that each periodic wave is embedded in a family of neighboring waves implies that 1
always belongs to the spectrum of S0 (1/ω, 0). It is then crucial in the analysis of linear stability
to determine how the eigenvalue 1 evolves when ξ is varied. Our goal is to prove that this piece
of information is precisely given by the averaged equations.
3.3. Algebraic interplay. To analyze the evolution of the spectrum
π π of
Sξ (1/ω, 0) when ξ is
small, we first stress the following classical identity, for any ξ ∈ − N
,N
Z t
2
(3.7)
∀(s, t) ∈ R , Sξ (t, s) = S0 (t, s) +
S0 (t, r) (Aξ (r) − A0 (r)) Sξ (r, s) dr .
s
In particular, expanding Aξ (·) as
ξ→0
Aξ (·) = A0 (·) + i ξ A(1) (·) + (i ξ)2 A(2) (·) + O(|ξ|3 ) ,
uniformly in time, for some A(1) (·), A(2) (·), we receive locally uniformly in time
ξ→0
Sξ (t, s) = S0 (t, s) + i ξ S (1) (t, s) + (i ξ)2 S (2) (t, s) + O(|ξ|3 ) ,
13
with
S (1) (t, s)
Z
t
S0 (t, r) A(1) (r) S0 (r, s) dr
=
s
(3.8)
S (2) (t, s)
Z
=
t
S0 (t, r) A
(2)
Z
(r) S0 (r, s) dr +
s
t
S0 (t, r) A(1) (r) S (1) (r, s) dr .
s
We have already encountered a similar expansion for Lξ := e−iξ· L eiξ· . Operators Lξ acting
on functions of period 1 are actually continuous Bloch symbols of the operator obtained by
extending L to functions defined on the full line. Since we won’t need this relation here, we
mostly use the fact that by expanding
Lξ = L + (iξ k) L(1) + (iξ k)2 L(2) + O(|ξ|3 ) .
we obtain operators L(j) that are practical shorthands to denote relations obtained by differentiating profile equation with respect to parameters. For instance, when a denotes a parameter
which is not k (but possibly ζ, to account for phase shifts), one receives
L∂a u = ∂a ω∂ζ u
(3.9)
while
L∂k u = −k∂k c ∂ζ u − L(1) ∂ζ u .
(3.10)
Proofs of our main results strongly rely on the interplay between Lξ and Aξ that allows to
use the previous relations. Explicitly to any function v : R → Cd of period 1, we associate
V v : R → (Cd )Z by
V v (j, t) = v(k (j − ct)) .
Observe that when v and w are one-periodic
Z 1/ω
1
hV w (s), V v (s)ids =
hw, vi .
kω
0
Note also that when v is one-periodic V v (·, t) is 1/ω-periodic in time and valued in N -periodic
sequences. In this case,
(3.11)
= −V Lξ v (t) − iξωV v (t)
d v
dt V (t)
−
d v
dt V (t)
+ Aξ (t)∗ V v (t) =
Aξ (t) V v (t)
∗
V Lξ v (t)
− iξωV v (t) .
In particular
∗
d
(hV w , V v i)(t) = hV Lξ w (t), V v (t)i − hV w (t), V Lξ v (t)i ,
dt
which turns to be crucial to transport duality relations. Obviously by expanding in ξ the
foregoing relations we obtain corresponding relations between operators A(j) and operators L(j) ,
for instance
d v
A0 (t) V v (t) = −V L v (t)
dt V (t) −
(3.12)
(1) −c) v
− A(1) (t) V v (t) = −k V (L
(3.13)
(2)
− A(2) (t) V v (t) = −k 2 V L
d v
dt V (t)
(t)
v (t)
∗
+ A0 (t)∗ V v (t) = V L v (t) .
Mark that from (3.9) and (3.11)
∀(t, s) ∈ R2 , V ∂ζ u (t) = S0 (t, s) V ∂ζ u (s) .
In particular, V ∂ζ u (0) ∈ ker(S0 (1/ω, 0) − Id). More generally (3.9) and (3.11) provide as many
members of the generalized kernel of S0 (1/ω, 0) − Id as the expected dimension of the family of
periodic traveling waves (counted up to translation). Regardless of the version of the equations
14
that we are considering, assumption A may be reformulated as the fact that these elements form
a basis of this generalized kernel. Moreover (3.10), (3.11) and (3.8) yield
(3.14) ∀(t, s) ∈ R2 , V ∂k u (t) = S0 (t, s) V ∂k u (s) − (t − s)∂k ω V ∂ζ u (s) +
1 (1)
S (t, s)V ∂ζ u (s) .
k
4. Spectral validation of averaged equations
4.1. Reaction-diffusion case. Firstly we analyze the linearization of (2.3) about U (·) the
periodic traveling wave associated to uk , with k = 1/N . The linearized evolution obeys
d
(∀t , dt
V (t) = A(t) V (t)) where
A(·) = µ (T − 2Id + T−1 )) + df (U (·))
with usual identification of functions with their local action on sequences, that is, here,
∀(t, j) , [df (U (t))V ]j = df (U j (t)) Vj .
Explicitly here
1
and A(2) (·) ≡ µ (T + T−1 ) .
2
Now our main goal is to prove the following result that validates (2.10) and (2.14) at the
spectral level.
A(1) (·) ≡ µ (T − T−1 )
Theorem 4.1. Assuming that condition A holds in a neighborhood of uk , there exist positive 0
and ξ0 ∈ (0, π/N ) and an analytic curve λ : [−ξ0 , ξ0 ] → C such that
∀ξ ∈ [−ξ0 , ξ0 ] , σ(Sξ (1/ω k , 0)) ∩ B(1, 0 ) = {λ(ξ)} .
Moreover, for ξ ∈ [−ξ0 , ξ0 ], eigenvalue λ(ξ) of Sξ (1/ω k , 0) is simple and
(4.1)
λ(ξ)
ξ→0
=
1
e ωk (
ikξ∂k ω k + (ikξ)2 d(k))
+ O(|ξ|3 ) ,
where d(k) is given by (2.13) under normalization N.
The above theorem has many direct implications. For instance, we readily see that d(k) > 0
yields a linear exponential instability caused by side-band perturbations, that is, by perturbations with arbitrary small — but non zero — spatial Floquet exponents. It also proves as
expected that the single linear group velocity is indeed ∂k ω k .
Proof. Assumption A implies that 1 is a simple eigenvalue of S0 (1/ω, 0) so that the only point
that does not follow from classical results on regular perturbations of simple eigenvalues is
expansion (4.1). To identify coefficients we expand
ξ→0
λ(ξ) = 1 + i ξ λ(1) + (i ξ)2 λ(2) + O(|ξ|3 ) ,
and a corresponding eigenvector ϕξ , which also depends analytically on ξ, as
ϕξ
ξ→0
= V ∂ζ u (0) + iξ ϕ(1) + (iξ)2 ϕ(2) + O(|ξ|3 ) .
Setting p = 1/ω k for writing convenience, we now expand relation Sξ (p, 0)ϕξ = λ(ξ)ϕξ .
At first order in iξ we receive
(4.2)
(S0 (p, 0) − Id) ϕ(1) = λ(1) V ∂ζ u (0) − S (1) (p, 0)V ∂ζ u (0) .
Hence
(1)
(S0 (p, 0) − Id) (ϕ
− kV
∂k u
(0)) =
λ
(1)
− kp∂k ω
k
V ∂ζ u (0) .
Since 1 is a simple eigenvalue of S0 (p, 0), V ∂ζ u (0) does not belong to the image of S0 (p, 0) − Id.
From this we infer the desired
λ(1) = kp∂k ω k
and
ϕ(1) − kV ∂k u (0) ∈ ker(S0 (p, 0) − Id) = CV ∂ζ u (0) .
15
By expanding now at the second order in iξ we obtain
(4.3) (S0 (p, 0) − Id) ϕ(2) = λ(2) V ∂ζ u (0) + kp∂k ω k ϕ(1) − S (1) (p, 0)ϕ(1) − S (2) (p, 0)V ∂ζ u (0) .
ad
∗
To proceed we also need to involve uad
k defined in Subsection 2.1, that is uk spans ker L and
k
2
huad
k , ∂ζ u i = 1 (in the Lper ([0, 1]) sense). From (3.11) we deduce that
∀(t, s) ,
S0 (s, t)∗ V ad (t) = V ad (s) ,
ad
thus that V uk (0) ∈ ker(S0 (p, 0)∗ − Id) and from (3.12) that
Z
1 p uad
1
1
uad
∂ζ u
k
k
∀t , hV (t), V
hV k (s), V ∂ζ u (s)i ds = huad
(t)i =
.
k , ∂ζ u i =
p 0
k
k
Since −kp∂k ω k V ∂ζ u (0) + S (1) (p, 0)V ∂ζ u (0) belongs to the range of S0 (p, 0) − Id this yields
1 (2)
kλ
ad
= hV uk (0), −kp∂k ω k V ∂k u (0) + kS (1) (p, 0)V ∂k u (0) + S (2) (p, 0)V ∂ζ u (0)i .
Now with (3.8) we compute
ad
hV uk (0) , kS (1) (p, 0)V ∂k u (0) + S (2) (p, 0)V ∂ζ u (0)i
Z p
ad
hV uk (s), A(1) (s)[kS0 (s, 0)V ∂k u (0) + S (1) (s, 0)V ∂ζ u (0)] + A(2) (s)V ∂ζ u (s)i ds
=
Z0 p
ad
=
hV uk (s), A(1) (s)[k s ∂k ω k V ∂ζ u (s) + kV ∂k u (s)] + A(2) (s)V ∂ζ u (s)i ds
Z0 p
ad
(1)
(2)
(1)
=
hV uk (s), k 2 s ∂k ω k V (L −c)∂ζ u (s) + k 2 V (L −c)∂k u (s)] + k 2 V L ∂ζ u (s)i ds .
0
(L(1) − c)∂
Now, since
ζ u = ∂k ω∂ζ u − L∂k u, by using (3.13) and normalization N and integrating
by part, we observe that
Z p
1 1 2 2
uad
uad
2
k
k 2
2
(L(1) −c)∂ζ u
k
∂k u
k
k
hV (s), k s ∂k ω V
k p (∂k ω ) + k p∂k ω hV (0), V
(s)i ds =
(0)i
k 2
0
which yields
λ(2) = pk 2 d(k) +
1
2
(pk∂k ω k )2
as desired.
4.2. General mixed case. Now our main goal is to unravel the role of (2.20) in the spectral
analysis of the linearization of (2.16).
Theorem 4.2. Assuming that condition A holds in a neighborhood of uk,M , there exist positive
0 and ξ0 ∈ (0, π/N ) and continuous curves λα : [−ξ0 , ξ0 ] → C, α = 1, · · · , d1 + 1, such that
∀ξ ∈ [−ξ0 , ξ0 ] , σ(Sξ (1/ω k,M , 0)) ∩ B(1, 0 ) = {λ1 (ξ), · · · , λd1 +1 (ξ)} .
Moreover, for ξ ∈ [−ξ0 , ξ0 ] and α = 1, · · · , d1 + 1
(4.4)
λα (ξ)
ξ→0
=
1
e ωk,M
ikξ aα
+ O(|ξ|2 ) ,
where a1 , · · · , ad1 +1 are the characteristic speeds of the Whitham system (2.20) linearized about
parameters (k, M).
In contrast with what happens in the reaction-diffusion case, that is averaged to a scalar equation, the above first-order expansion already provides an instability criterion. If the Whitham
system (2.20) is not weakly hyperbolic at (k, M), that is, if some of the aα are not real, then the
corresponding wave is linearly exponentially instable to side-band perturbations. In contrast,
when weak hyperbolicity is met, then the above theorem proves that those aα do give the d1 + 1
linear group velocities, a non trivial fact.
16
Proof. Let us denote by e1 , · · · , ed1 both the d1 first members of the canonical basis of Cd and
the constant functions with corresponding values. As a consequence of assumption A, there
exists uad belonging to the generalized kernel of L∗ , orthogonal to ∂M1 u, · · · , ∂Md1 u and such
that huad , ∂ζ ui = 1. Moreover, by appealing to Subsection 3.3 as in the reaction-diffusion case,
ad
∂
u
one obtains that (V ∂ζ u , V ∂M1 u , · · · , V Md1 ) and (kV u , kV e1 , · · · , kV ed1 ) form dual bases of
the generalized kernels of S0 (1/ω k,M , 0) − Id and S0 (1/ω k,M , 0)∗ − Id.
By Kato’s perturbation method [Kat76, pp. 99-100] we may continue these bases as dual
bases (q0 (ξ), · · · , qd1 (ξ)) and (˜
q0 (ξ), · · · , q˜d1 (ξ)) of the generalized eigenspaces of Sξ (1/ω k,M , 0)
∗
k,M
, 0) associated with their spectra in B(1, 0 ) as long as |ξ| ≤ ξ0 provided ξ0 and
and Sξ (1/ω
0 are small enough. Then, for |ξ| ≤ ξ0 , the spectrum of Sξ (1/ω k,M , 0) − Id in B(0, 0 ) is the
spectrum of the matrix
hD
Ei
.
Ωξ =
q˜α (ξ), Sξ (1/ω k,M , 0) − Id qβ (ξ)
0≤α,β≤d1
Observe that

Ω0
0 p∂M1 ω
···
 ..
= .
0d1 ×d1
0
p∂Md1 ω



where p = 1/ω k,M . Expanding
Ωξ = Ω0 + ikξ Ω(1) + (ikξ)2 Ω(2) + O(|ξ|3 ) ,
we also obtain since
∂ζ u
1 (1)
(0)
k S (p, 0) V
= −(S0 (p, 0) − Id) V ∂k u (0) + p ∂k ω k V ∂ζ u (0)
that
Ω(1)

∗ ···
0

=  ..
. ∗
0
∗



.

˜ ξ defined by
The upshot is that the matrix Ω
˜ ξ :=
Ω
1 −1
Σ Ωξ Σξ
ikξ ξ
with
Σξ
(ikξ)−1 O1×d1
=
0d1 ×1 Id1 ×d1
depends analytically on ξ. The heuristics behind this transformation is that the change of basis
transforms a phase-like coordinate to a wavenumber-like coordinate as in the derivation of the
Whitham system (2.20) while the division by ikξ changes eigenvalues in velocities.
˜ 0 . We already know that
Now the point is to identify Ω
(Ω(1) )α,β (Ω0 )α,β
˜
.
Ω0 =
(Ω(2) )α,β (Ω(1) )α,β 0≤α,β≤d
1
and, for β = 1, · · · , d1 , (Ω0 )1,β = p∂Mβ ω. First
S0 (p, 0) − Id ∂ξ q0 (0)i + k1 h˜
q0 (0), S (1) (p, 0)q0 (0)i
= p∂k ω + ik1 h˜
q0 (0), S0 (p, 0) − Id (∂ξ q0 (0) − ikV ∂k u (0))i .
(Ω(1) )0,0 =
1
q0 (0),
ik h˜
Our claim is that the last term vanishes. Indeed since (S0 (p, 0) − Id)q0 (0) = 0 and, for |ξ| ≤ ξ0 ,
(Sξ (p, 0) − Id)q0 (ξ) ∈ Span({q0 (ξ), · · · , qd1 (ξ)})
we conclude
1
ik (S0 (p, 0)
− Id)∂ξ q0 (0) +
1 (1)
k S (p, 0)q0 (0)
17
∈ Span({q0 (0), · · · , qd1 (0)})
hence
S0 (p, 0) − Id (∂ξ q0 (0) − ikV ∂k u (0)) ∈ Span({q0 (0), · · · , qd1 (0)}) .
This implies
∂ξ q0 (0) − ikV ∂k u (0) ∈ Span({q0 (0), · · · , qd1 (0)})
which in turn yields
S0 (p, 0) − Id (∂ξ q0 (0) − ikV ∂k u (0)) ∈ Span({q1 (0), · · · , qd1 (0)}) ,
proving the cancellation. To make the remaining computations easier we pick (z1 , · · · , zd1 ) such
that
X
∂ξ q0 (0) − ikV ∂k u (0) =
zα qα (0)
1≤α≤d1
Pd1
and replace (q0 (ξ), q˜1 (ξ), · · · , q˜d1 (ξ)) with (q0 (ξ) − ξ α=1 zα qα (ξ), q˜1 (ξ) + ξz1 q˜0 (ξ), · · · , q˜d1 (ξ) +
ξzd1 q˜0 (ξ)). This ensures ∂ξ q0 (0) = ikV ∂k u (0).
With this in hands we may also compute, for 1 ≤ α, β ≤ d1 ,
(Ω(1) )α,β = ik1 h∂ξ q˜α (0), S0 (p, 0) − Id qβ (0)i + k1 h˜
qα (0), S (1) (p, 0)qβ (0)i
=
p∂Mβ ω
˜α (0), q0 (0)i
ik h∂ξ q
= −
+
p∂Mβ ω
qα (0), ∂ξ q0 (0)i
ik h˜
1
qα (0), S (1) (p, 0)qβ (0)i
k h˜
+
1
qα (0), S (1) (p, 0)qβ (0)i
k h˜
= −p∂Mβ ωh˜
qα (0), V ∂k u (0)i +
1
qα (0), S (1) (p, 0)qβ (0)i .
k h˜
Now using that, for any (s, t),
S0 (t, s)V
∂ Mβ u
(s) = V
∂ Mβ u
(t) + (t − s)∂Mβ ωV ∂ζ u (t) ,
we go further to obtain
1
qα (0), S (1) (p, 0)qβ (0)i
k h˜
with
Z
Z
=
p
hV eα (t), A(1) (t)[V
∂ Mβ u
(t) + t∂Mβ ωV ∂ζ u (t)]i dt
0
p
hV eα (t), A(1) (t)V
∂ Mβ u
(t)i dt = p heα , (L(1) − c)∂Mβ ui
0
1
Z
ηeα · df (u(ζ))∂Mβ u(ζ)dζ
= p
0
= −p ∂Mβ Fα (k, M )
(L(1)
and, by using
− c)∂ζ u = ∂k ω∂ζ u − L∂k u and integrating by part,
Z p
Z p
eα
(1)
∂ζ u
t hV (t), A (t)V
t hV eα (t), V L∂k u (t)]i dt
(t)]i dt = −k
0
0
Z p
d
= k
t hV eα (t), dt
− A0 (t)) V ∂k u (t)]i dt
0
= k p hV eα (0), V ∂k u (0)i .
Hence, for 1 ≤ α, β ≤ d1 ,
(Ω(1) )α,β = −p ∂Mβ Fα (k, M ) .
Lastly we compute, for 1 ≤ α ≤ d1 ,
(Ω(2) )α,0 =
1
h∂ q˜ (0),
(ik)2 ξ α
S0 (p, 0) − Id ∂ξ q0 (0)i +
+ ik12 h∂ξ q˜α (0), S (1) (p, 0) q0 (0)i +
18
1
h˜
q (0), S (1) (p, 0)∂ξ q0 (0)i
ik2 α
1
h˜
q (0), S (2) (p, 0) q0 (0)i .
k2 α
On one hand, integrating by parts,
1
ik2
Z p
h∂ξ q˜α (0), S0 (p, t)A(1) (t)V ∂ζ u (t)i dt
h∂ξ q˜α (0), S (1) (p, 0) q0 (0)i = ik12
0
Z p
Z p
∂ζ u
1
1
d
= ik
h∂ξ q˜α (0), S0 (p, t)∂k ω V
(t)i dt + ik
h∂ξ q˜α (0), S0 (p, t) dt
− A0 (t) V ∂k u (t)i dt
0
i
h0
= ik1 p∂k ω h∂ξ q˜α (0), V ∂ζ u (0)i − ik1 h∂ξ q˜α (0), S0 (p, 0) − Id ∂ξ q0 (0)i .
On the other hand, using (3.14),
1
k2
h˜
qα (0), [ 1i S (1) (p, 0)∂ξ q0 (0) + S (2) (p, 0) q0 (0)]i
Z p
h
i
hV eα (t), A(1) (t) V ∂k u (t) + t∂k ω V ∂ζ u (t) i dt +
=
0
= p heα , (L(1) − c)∂k u + L(2) ∂ζ ui + p ∂k ω
1
k
Z
p
hV eα (t), A(2) (t) V ∂ζ u (t)i dt
0
1
qα (0), ∂ξ q0 (0)i .
ik h˜
Hence
(Ω(2) )α,0 = p heα , (L(1) − c)∂k u + L(2) ∂ζ ui = −p ∂k Fα (k, M )
so that as expected

∂M1 ω
∂k ω
˜0 = p 
Ω
···
∂Md1 ω

.
−∂k Fα (k, M ) −∂M1 Fα (k, M ) · · ·
−∂Md1 Fα (k, M )
4.3. Hamiltonian case. Now we turn our attention to the role of (2.26) in the spectral analysis
of the linearization of (2.21). Our analysis hereafter follows closely the proof of the foregoing
subsection. The only difference stems from the presence of one extra ”hidden” conservation law.
Theorem 4.3. Assuming that condition A holds in a neighborhood of uk,M,E , there exist positive
0 and ξ0 ∈ (0, π/N ) and continuous curves λα : [−ξ0 , ξ0 ] → C, α = 1, · · · , d + 2, such that
∀ξ ∈ [−ξ0 , ξ0 ] , σ(Sξ (1/ω k,M , 0)) ∩ B(1, 0 ) = {λ1 (ξ), · · · , λd+2 (ξ)} .
Moreover, for ξ ∈ [−ξ0 , ξ0 ] and α = 1, · · · , d + 2
(4.5)
λα (ξ)
ξ→0
=
1
e ωk,M
ikξ aα
+ O(|ξ|2 ) ,
where a1 , · · · , ad+2 are the characteristic speeds of the Whitham system (2.26) linearized about
parameters (k, M, E).
Proof. As in the previous subsection, perturbatively we may obtain dual bases (q0 (ξ), · · · , qd+1 (ξ))
and (˜
q0 (ξ), · · · , q˜d+1 (ξ)) of the generalized eigenspaces of Sξ (1/ω k,M,E , 0) and Sξ (1/ω k,M,E , 0)∗
associated with their spectra in B(1, 0 ) as long as |ξ| ≤ ξ0 provided ξ0 and 0 are small enough.
Here however we start from
(q0 (0), · · · , qd+1 (0)) = (V ∂ζ u , V ∂M1 u , · · · , V ∂Md u , V ∂E u )
and
ad
(˜
q0 (0), · · · , q˜d+1 (0)) = (kV u , kV e1 , · · · , kV ed , kV δk H[u] ) ,
where uad belongs to the generalized kernel of L∗ , is orthogonal to ∂M1 u, · · · , ∂Md u, ∂E u and
satisfies huad , ∂ζ ui = 1. Then, for |ξ| ≤ ξ0 , the spectrum of Sξ (1/ω k,M,E , 0) − Id in B(0, 0 ) is
the spectrum of the matrix
hD
Ei
Ωξ =
q˜α (ξ), Sξ (1/ω k,M,E , 0) − Id qβ (ξ)
.
0≤α,β≤d+1
19
˜ ξ defined by
The cancellation discussed in the previous subsection shows that the matrix Ω
(ikξ)−1 O1×d+1
˜ ξ := 1 Σ−1 Ωξ Σξ
Ω
with
Σξ =
ξ
0d+1×1 Id+1×d+1
ikξ
˜ 0 with our averaged system, here (2.26).
depends analytically on ξ. The point is again to connect Ω
Manipulations as above already yield


∂k ω
∂M1 ω
···
∂Md ω
∂E ω




˜

,
−B∂
F
−B∂
F
·
·
·
−B∂
F
−B∂
F
E
M1
Md
k
Ω0 = p 



1
1
1
1
(2) )
(1) )
(1) )
(1) )
(Ω
(Ω
·
·
·
(Ω
(Ω
d+1,0 p
d+1,1
d+1,d p
d+1,d+1
p
p
where p = 1/ω k,M,E , for α = 1, · · · , d + 1,
1
h∂ q˜ (0),
(ik)2 ξ d+1
(Ω(2) )d+1,0 =
S0 (p, 0) − Id ∂ξ q0 (0)i +
+ ik12 h∂ξ q˜d+1 (0), S (1) (p, 0) q0 (0)i +
1
h˜
q (0), S (1) (p, 0)∂ξ q0 (0)i
ik2 d+1
1
h˜
q (0), S (2) (p, 0) q0 (0)i ,
k2 d+1
and, for β = 1, · · · , d + 1,
(Ω(1) )d+1,β =
1
˜d+1 (0),
ik h∂ξ q
S0 (p, 0) − Id qβ (0)i +
Moreover we may enforce ∂ξ q0 (0) = ik∂k u.
For β = 1, · · · , d + 1, setting
(
Mβ
aβ =
E
1
qd+1 (0), S (1) (p, 0)qβ (0)i .
k h˜
if 1 ≤ β ≤ d
if β = d + 1
and proceeding as in the previous subsection we derive
(Ω(1) )d+1,β =
p
ik
∂aβ ω h∂ξ q˜d+1 (0), q0 (0)i + p hδk H[u], (L(1) − c)∂aβ ui
Z p
d
− A0 (t) V ∂k u (t)]i dt .
+ k ∂aβ ω
t hV δk H[u] (t), dt
0
Now using (2.24) and (2.25) we also obtain
(1)
p hδk H[u], (L
Z
− c)∂aβ ui = p ∂aβ S + k ∂aβ ω
p
hV δk H[u] (t), V ∂k u (t)]i dt .
0
From this stems as expected
(Ω(1) )d+1,β =
p
ik
∂aβ ω [h∂ξ q˜d+1 (0), q0 (0)i + h˜
qd+1 (0), ∂ξ q0 (0)i] + p ∂aβ S = p ∂aβ S .
Lastly, on one hand, we still have
1
h∂ q˜ (0) ,
ik2 ξ d+1
S (1) (p, 0) q0 (0)i
h
1
= ik p ∂k ω h∂ξ q˜d+1 (0), q0 (0)i −
1
˜d+1 (0),
ik h∂ξ q
i
S0 (p, 0) − Id ∂ξ q0 (0)i .
On the other hand,
1
k2
h˜
qα (0), [ 1i S (1) (p, 0)∂ξ q0 (0) + S (2) (p, 0) q0 (0)]i
Z p
Z p
h
i
eα
(1)
∂k u
∂ζ u
1
=
hV (t), A (t) V
(t) + t∂k ω V
(t) i dt + k
hV eα (t), A(2) (t) V ∂ζ u (t)i dt
0
0
Z p
d
= p heα , (L(1) − c)∂k u + L(2) ∂ζ ui + k ∂k ω
t hV δk H[u] (t), dt
− A0 (t) V ∂k u (t)]i dt
0
= p ∂k S + p ∂k ω
1
qα (0), ∂ξ q0 (0)i .
ik h˜
20
Hence as expected (Ω(2) )d+1,0 = p ∂k S.
This achieves the proof of

∂k ω
∂M1 ω


˜ 0 = p  −B∂k F −B∂M1 F
Ω


∂k S
∂M1 S
···
∂Md ω
···
−B∂Md F
···
∂M d S
that yields the theorem.
∂E ω



−B∂E F 


∂E S
5. Conclusions and Remarks
We have proved that the slow modulation ansatz captures accurately the essential spectral
features of the space-time low2 Floquet-multiplier evolution linearized about a periodic wave
whose period belongs to the lattice. As already mentioned this opens at least three classes of
questions.
In many cases one also expects that in the large-time the dynamics is effectively reduced to
a slow modulation evolution. Our analysis provides the spectral background — including, as
appears from our proofs, expansion of critical eigenfunctions — for a proof of such claims, either
in the linear regime as, for continuous systems, in [Rod] for the Korteweg–de Vries equation,
or in the complete nonlinear regime, as in [JNRZ14] for general parabolic systems of partial
differential equations.
With this in mind, one expects to be able to identify on averaged systems the key features
of the discrete dynamics. It is then natural to ask if one may design discrete systems leading
to desired modulation properties, either with applications to the analysis of numerical schemes
where the goal is to preserve the slow modulation asymptotics at the discrete level, or in the
effective conception of smart materials, a fast-growing field of investigation.
At last, even for the direct problem at the spectral level, remains the technical challenge of
dealing with waves of general period, a seemingly quasi-periodic problem.
Acknowledgement: The main part of this work was carried out during the 3-months stay
of B.K. in Lyon on 2013. B.K. would like to thank Sylvie Benzoni-Gavage for the kind invitation
that was at the origin of this stay, and acknowledge the hospitality of Institut Camille Jordan
at the Universit´e Lyon 1 and the financial support of the CMIRA ACCUEIL DOC program of
the region Rhˆ
one-Alpes.
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˘ ra Kabil, Institute of Applied Analysis and Numerical Simulation, Pfaffenwaldring 57,
Bug
70569 Stuttgart, University of Stuttgart, Germany
E-mail address: [email protected]
´ Kaliffe,
´ de Lyon, CNRS UMR 5208, Universite
´ Lyon 1, INRIA EP
L.Miguel Rodrigues, Universite
Institut Camille Jordan, 43 bd 11 novembre 1918; F-69622 Villeurbanne cedex
E-mail address: [email protected]
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